When is a short exact sequence of three word-hyperbolic groups, Mahan Mj (formerly Mitra) has shown that the inclusion map from to extends continuously to a map between the Gromov boundaries of and . This boundary map is known as the Cannon–Thurston map. In this context, Mj associates to every point in the Gromov boundary of an “ending lamination” on which consists of pairs of distinct points in the boundary of . We prove that for each such , the quotient of the Gromov boundary of by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich and Lustig and Dowdall, Kapovich and Taylor, who prove that in the case where is a free group and is a convex cocompact purely atoroidal subgroup of , one can identify the resultant quotient space with a certain –tree in the boundary of Culler and Vogtmann’s Outer space.
"Trees, dendrites and the Cannon–Thurston map." Algebr. Geom. Topol. 20 (6) 3083 - 3126, 2020. https://doi.org/10.2140/agt.2020.20.3083